Finite Group Databasebeta

← #27283224

Subgroups — #27283224

Order 13568.

742 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20274695C211yesmaximal, characteristic, complemented
2C2×#16810360C212yesmaximal, complemented
3C2×#16810357C212yesmaximal
4C4×#13133124C211yesmaximal, characteristic, complemented
5C53×#2850C211yesmaximal, characteristic
6C22×C4×Dic53—21no—
7C2×C4⋊#88322—21no—
8C2×C22⋊#88322C2212yes—
9C2×#13133124—21no—
10C2×#13133124C2212yes—
11#16810378—21no—
12#16810360—24no—
13#16810357—24no—
14C2×#124×C53C411yescharacteristic, complemented
15C2×C22⋊Q8×C53C2211yescharacteristic
16C2×#121×C53C2211yescharacteristic
17C22×C42×C53C2211yescharacteristic
18C53×#391C411yescharacteristic, complemented
19C53×#390C411yescharacteristic, complemented
20C53×#388—21no—
21C53×#388—21no—
22C53×#386—21no—
23C53×#386—21no—
24C53×#382C411yescharacteristic, complemented
25C23×Dic53—22no—
26C22×#88322—22no—
27C2×C4×Dic53—24no—
28C2×C4×Dic53—42no—
29C2×C4×Dic53—42no—
30C4⋊#88322—42no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133124—24no—
34#13133124—22no—
35#13133124D414yes—
36C22×Q8×C53C2×C411yescharacteristic
37C22×D4×C53—21no—
38C22×D4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C23×C4×C53C2311yescharacteristic
42#124×C53—42no—
43C22⋊Q8×C53—22no—
44C22⋊Q8×C53—22no—
45#121×C53—22no—
46#121×C53—22no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C42×C53—42no—
57C2×C42×C53C2×C411yescharacteristic
58C2×C42×C53C2×C411yescharacteristic
59C2×C42×C53D411yescharacteristic
60C2×C42×C53D411yescharacteristic
61#95×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64#95×C53—21no—
65C24×C53—21no—
66C24×C53C2×D411yescharacteristic
67C2×Q8×C53—22no—
68C2×Q8×C53—41no—
69C2×Q8×C53—41no—
70C2×D4×C53—22no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—42no—
74C2×D4×C53—42no—
75C2×D4×C53—41no—
76C2×D4×C53—41no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53—22no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53C22⋊C411yescharacteristic
90C22×C4×C53C22⋊C411yescharacteristic
91C22×C4×C53C22⋊C411yescharacteristic
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22×C411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D412yes—
96C22×C4×C53#4112yes—
97C22×Dic53—24no—
98C22×Dic53—24no—
99C22×Dic53—24no—
100C22×Dic53—42no—
101C22×Dic53—42no—
102C22×Dic53—42no—
103C22×Dic53—42no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C22⋊C4×C53—42no—
109C22⋊C4×C53—42no—
110C22⋊C4×C53—42no—
111C22⋊C4×C53—42no—
112C22⋊C4×C53—42no—
113C22⋊C4×C53—41no—
114C22⋊C4×C53—41no—
115C42×C53—22no—
116C42×C53—22no—
117C42×C53—41no—
118C42×C53—41no—
119C4×Dic53—48no—
120C2×#88322—42no—
121C2×#88322—42no—
122C2×#88322—42no—
123C23×C53—21no—
124C23×C53—21no—
125C23×C53—21no—
126C23×C53—21no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53C2×C22⋊C411yescharacteristic
135C23×C53#12112yes—
136C23×C53#13512yes—
137C23×C53AGL(1,ℤ/8)11yescharacteristic
138C23×C53#13711yescharacteristic
139Q8×C53—82no—
140D4×C53—82no—
141D4×C53—82no—
142D4×C53—82no—
143C2×C4×C53—24no—
144C2×C4×C53—24no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—44no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53#9912yes—
171C2×C4×C53#10412yes—
172C2×C4×C53#10412yes—
173C2×C4×C53C4×D412yes—
174C2×C4×C53#12411yescharacteristic
175C2×C4×C53#12711yescharacteristic
176C2×Dic53—28no—
177C2×Dic53—44no—
178C2×Dic53—44no—
179C2×Dic53—44no—
180C2×Dic53—44no—
181C2×Dic53—44no—
182C2×Dic53—44no—
183#88322—82no—
184#9713—531nomaximal, Sylow
185C22×C53—22no—
186C22×C53—22no—
187C22×C53—22no—
188C22×C53—22no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53—81no—
196C22×C53—81no—
197C22×C53—81no—
198C22×C53—81no—
199C22×C53—81no—
200C22×C53#39011yescharacteristic
201C22×C53C2×#9911yescharacteristic
202C22×C53#41011yescharacteristic
203C22×C53#41611yescharacteristic
204C22×C53#41711yescharacteristic
205C22×C53C2×#10411yescharacteristic
206C22×C53C2×#10411yescharacteristic
207C22×C53#46312yes—
208C22×C53#46412yes—
209C4×C53—21no—
210C4×C53—22no—
211C4×C53—21no—
212C4×C53—22no—
213C4×C53—22no—
214C4×C53—22no—
215C4×C53—41no—
216C4×C53—81no—
217C4×C53—81no—
218C4×C53—81no—
219Dic53—48no—
220#2850D5311yescharacteristic
221#1797—531no—
222C2×#328—532no—
223C2×#327—532no—
224C4×#95—531no—
225C2×C53—22no—
226C2×C53—22no—
227C2×C53—161no—
228C2×C53#150011yescharacteristic
229C2×C53#150111yescharacteristic
230C2×C53#150411yescharacteristic
231C2×C53#150911yescharacteristic
232C2×C53#196211yescharacteristic
233C2×C53#196411yescharacteristic
234C2×C53#198311yescharacteristic
235C2×#124Dic5311yescharacteristic, complemented
236C2×C22⋊Q8C2×D5311yescharacteristic
237C2×#121C2×D5311yescharacteristic
238C22×C42—1061no—
239C22×C42C2×D5311yescharacteristic
240C2×C4⋊C8—1061no—
241C2×C22⋊C8—532no—
242#391Dic5311yescharacteristic, complemented
243#390Dic5311yescharacteristic, complemented
244#388—21no—
245#388—21no—
246#386—21no—
247#386—21no—
248#382Dic5311yescharacteristic, complemented
249C2×#95—532no—
250C2×#95—1061no—
251#336—1061no—
252#328—1064no—
253#327—1064no—
254C53#971311yescharacteristic, complemented, Sylow
255C22×Q8C2×Dic5311yescharacteristic
256C22×D4—21no—
257C22×D4C2×Dic5311yescharacteristic
258C23×C4—1062no—
259C23×C4C22⋊D5311yescharacteristic
260C23×C4C22⋊D5311yescharacteristic
261C23×C4C22×D5311yescharacteristic
262C22×C8—1062no—
263#124—42no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266#121—22no—
267#121—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C2×Dic5311yescharacteristic
270C2×C4⋊C4C2×Dic5311yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C42—42no—
278C2×C42—1064no—
279C2×C42—2122no—
280C2×C42—2122no—
281C2×C42C2×Dic5311yescharacteristic
282C2×C42C2×Dic5311yescharacteristic
283C2×C42C22⋊D5311yescharacteristic
284C2×C42C22⋊D5311yescharacteristic
285C4⋊C8—2122no—
286C22⋊C8—1062no—
287C22⋊C8—1062no—
288#95—21no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—534no—
293#95—1064no—
294#95—1062no—
295C24—21no—
296C24C2×C22⋊D5311yescharacteristic
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—1064no—
320C22×C4—1064no—
321C22×C4—1064no—
322C22×C4—2122no—
323C22×C4—2122no—
324C22×C4—2122no—
325C22×C4—2122no—
326C22×C4C22⋊Dic5311yescharacteristic
327C22×C4C22⋊Dic5311yescharacteristic
328C22×C4C22⋊Dic5311yescharacteristic
329C22×C4C22⋊Dic5311yescharacteristic
330C22×C4D4×D5312yes—
331C22×C4#1171994712yes—
332C22×C4C22×Dic5311yescharacteristic
333C22×C4C2×C22⋊D5311yescharacteristic
334C2×C8—2122no—
335C2×C8—2122no—
336C2×C8—2122no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—2128no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23C2×C22⋊Dic5311yescharacteristic
365C23#1313318812yes—
366C23#1313323712yes—
367C23#1313324211yescharacteristic
368C23#1313324711yescharacteristic
369Q8—82no—
370D4—82no—
371D4—82no—
372D4—82no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—1068no—
401C2×C4—2124no—
402C2×C4—2124no—
403C2×C4—2124no—
404C2×C4—2124no—
405C2×C4—2124no—
406C2×C4—2124no—
407C2×C4#1313313212yes—
408C2×C4#1313314412yes—
409C2×C4#1313314412yes—
410C2×C4D4×Dic5312yes—
411C2×C4#1313320411yescharacteristic
412C2×C4#1313321311yescharacteristic
413C8—4242no—
414C22—22no—
415C22—22no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—22no—
421C22—22no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#1681052611yescharacteristic
430C22C2×#1313313211yescharacteristic
431C22#1681059711yescharacteristic
432C22#1681061911yescharacteristic
433C22#1681062411yescharacteristic
434C22C2×#1313314411yescharacteristic
435C22C2×#1313314411yescharacteristic
436C22#1681083512yes—
437C22#1681084212yes—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—21no—
442C4—22no—
443C4—21no—
444C4—41no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—2128no—
449C2—22no—
450C2—22no—
451C2—161no—
452C2#2027320111yescharacteristic
453C2#2027320611yescharacteristic
454C2#2027321911yescharacteristic
455C2#2027324011yescharacteristic
456C2#2027554411yescharacteristic
457C2#2027555211yescharacteristic
458C2#2027564811yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2850C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#2850D53
hypocenterC53#9713
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #20274695C2semidirect
2. C2×#16810360C2semidirect
4. C4×#13133124C2semidirect
14. C2×#124×C53C4semidirect
18. C53×#391C4semidirect
19. C53×#390C4semidirect
24. C53×#382C4semidirect
235. C2×#124Dic53semidirect
242. #391Dic53semidirect
243. #390Dic53semidirect
248. #382Dic53semidirect
254. C53#9713semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202746952
C2×#168103602
C2×#168103572
C4×#131331242
C53×#28502
#971353